Studies On The Integrability And Symmetries Of The Nonlinear Coupled Jaulent-Miodek Equation

Soliton theory, as an important aspect of the nonlinear science, has a wide range of applications in natural scientific fields, such as nonlinear optics, hydrodynamics, biology and oceanography, etc. As an important research area, the nonlinear partial differential equations have attracted more and more mathematicians and physicists’ attentions to study the exact solutions of the nonlinear system. Due to the complexity of the nonlinear systems, to find the solutions of the nonlinear systems is a very difficult problem. Fortunately, people have proposed a series of effective methods to construct the exact solutions of the nonlinear system such as the inverse scattering method, the bilinear form, variable separation approach, symmetry reduction, Backlund transformation, Painlevé analysis, the Darboux transformation, classical and nonclassical Lie group method and so on.In this thesis, we firstly prove the Painlevé integrability or nonintegrability and exact solutions of a special coupled Jaulent-Miodek equation by the WTC method, then take use of the classical Lie group method and the Clarkson-Kruskal direct method to obtain the symmetries and the reduction equations. The thesis is divided into the following three parts:In Chapter 1, we briefly introduce the history and research background of soliton theory. The general steps of the Painlevé analysis, classical Lie group method and Clarkson-Kruskal direct method are proposed and the purpose and significance of the thesis are also given out.In Chapter 2, we firstly study the Painlevé integrability of the coupled Jaulent-Miodek equation by the Kruskal’s simplification of WTC method. In addition, new exact solutions of the coupled Jaulent-Miodek equation are constructed by the Painlevé standard and nonstandard truncation expansion. We also give detailed structure of the exact solutions.In Chapter 3, the classical symmetry of the coupled Jaulent-Miodek equation is studied, which consists of a three-parameter group. And two different types of similarity variables and similarity solutions are obtained by the proper parameter selection. Secondly, we use Clarkson-Kruskal’s direct method for finding all possible similarity reductions of Jaulent-Miodek equation. Discussions and summary are given in the last chapter.